4.10.3 Matrices, SPM Paper 2 (Long Questions)


Question 8:
The inverse matrix of   ( 4 1 2 5 ) is t( 5 1 2 n ).
(a) Find the value of n and of t.
(b) Write the following simultaneous linear equations as matrix equation:
4xy = 7
2x + 5y = –2
Hence, using matrix method, calculate the value of x and of y.


Solution:
(a) t( 5 1 2 n )= ( 4 1 2 5 ) 1 = 1 ( 4 )( 5 )( 1 )( 2 ) ( 5 1 2 4 ) = 1 22 ( 5 1 2 4 ) t= 1 22 , n=4

(b) ( 4 1 2 5 )( x y )=( 7 2 )   ( x y )= 1 22 ( 5 1 2 4 )( 7 2 )   ( x y )= 1 22 ( ( 5 )( 7 )+1( 2 ) ( 2 )( 7 )+( 4 )( 2 ) )   ( x y )= 1 22 ( 352 148 )   ( x y )= 1 22 (  33 22 )   ( x y )=(   3 2 1 ) x= 3 2 ,  y=1


Question 9:
(a) Given  1 s ( 4 2 5 3 )( t 2 5 4 )=( 1 0 0 1 ), find the value of s and of t.

(b) Using matrices, calculate the value of x and of y that satisfy the following matrix equation:
( 4 2 5 3 )( x y )=( 1 2 )


Solution:
(a) 1 s ( t 2 5 4 )= ( 4 2 5 3 ) 1 = 1 ( 4 )( 3 )( 2 )( 5 ) ( 3 2 5 4 ) = 1 2 ( 3 2 5 4 ) s=2, t=3

(b) ( 4 2 5 3 )( x y )=( 1 2 )   ( x y )= 1 2 ( 3 2 5 4 )( 1 2 )   ( x y )= 1 2 ( ( 3 )( 1 )+( 2 )( 2 ) ( 5 )( 1 )+( 4 )( 2 ) )   ( x y )= 1 2 ( 1 3 )   ( x y )=( 1 2 3 2 ) x= 1 2 ,  y= 3 2

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